Integrability and regularity of 3D Euler and equations for uniformly rotating fluids

AbstractWe consider 3D Euler and Navier-Stokes equations describing dynamics of uniformly rotating fluids. Periodic boundary conditions are imposed, and the ratio of domain periods is assumed to be generic (nonresonant). We show that solutions of 3D Euler/Navier-Stokes equations can be decomposed as U(t, x1, x2, x3) = Ũ(t, x1, x2) + V(t, x1, x2, x3) + r where Ũ is a solution of the 2D Euler/Navier-Stokes system with vertically averaged initial data (axis of rotation is taken along the vertical 3). Here r is a remainder of order Ro12a where Roa = (H0U0(Щ0L20) is the anisotropic Rossby number (H0—height, L0—horizontal length scale, Щ0—angular velocity of background rotation, U0—horizontal velocity scale); Roa = (H0L0)Ro where H0L0 is the aspect ratio and Ro = U0(Щ0L0) is a Rossby number based on the horizontal length scale L0. The vector field V(t, x1, x2, x3) which is exactly solved in terms of 2D dynamics of vertically averaged fields is phase-locked to the phases 2Щ0t, τ1(t), and τ2(t). The last two are defined in terms of passively advected scalars by 2D turbulence. The phases τ1(t) and τ2(t) are associated with vertically averaged vertical vorticity curl Ū(t) · e3 and velocity Ū3(t); the last is weighted (in Fourier space) by a classical nonlocal wave operator. We show that 3D rotating turbulence decouples into phase turbulence for V(t, x1, x2, x3) and 2D turbulence for vertically averaged fields Ū(t, x1, x2) if the anisotropic Rossby number Roa is small. The mathematically rigorous control of the error r is used to prove existence on a long time interval T∗ of regular solutions to 3D Euler equations (T∗ → +∞, as Roa → 0); and global existence of regular solutions for 3D Navier-Stokes equations in the small anisotropic Rossby number case

Continuation for thin film hydrodynamics and related scalar problems

This chapter illustrates how to apply continuation techniques in the analysis of a particular class of nonlinear kinetic equations that describe the time evolution through transport equations for a single scalar field like a densities or interface profiles of various types. We first systematically introduce these equations as gradient dynamics combining mass-conserving and nonmass-conserving fluxes followed by a discussion of nonvariational amendmends and a brief introduction to their analysis by numerical continuation. The approach is first applied to a number of common examples of variational equations, namely, Allen-Cahn- and Cahn-Hilliard-type equations including certain thin-film equations for partially wetting liquids on homogeneous and heterogeneous substrates as well as Swift-Hohenberg and Phase-Field-Crystal equations. Second we consider nonvariational examples as the Kuramoto-Sivashinsky equation, convective Allen-Cahn and Cahn-Hilliard equations and thin-film equations describing stationary sliding drops and a transversal front instability in a dip-coating. Through the different examples we illustrate how to employ the numerical tools provided by the packages auto07p and pde2path to determine steady, stationary and time-periodic solutions in one and two dimensions and the resulting bifurcation diagrams. The incorporation of boundary conditions and integral side conditions is also discussed as well as problem-specific implementation issues

Global Well-posedness of an Inviscid Three-dimensional Pseudo-Hasegawa-Mima Model

The three-dimensional inviscid Hasegawa-Mima model is one of the fundamental models that describe plasma turbulence. The model also appears as a simplified reduced Rayleigh-B\'enard convection model. The mathematical analysis the Hasegawa-Mima equation is challenging due to the absence of any smoothing viscous terms, as well as to the presence of an analogue of the vortex stretching terms. In this paper, we introduce and study a model which is inspired by the inviscid Hasegawa-Mima model, which we call a pseudo-Hasegawa-Mima model. The introduced model is easier to investigate analytically than the original inviscid Hasegawa-Mima model, as it has a nicer mathematical structure. The resemblance between this model and the Euler equations of inviscid incompressible fluids inspired us to adapt the techniques and ideas introduced for the two-dimensional and the three-dimensional Euler equations to prove the global existence and uniqueness of solutions for our model. Moreover, we prove the continuous dependence on initial data of solutions for the pseudo-Hasegawa-Mima model. These are the first results on existence and uniqueness of solutions for a model that is related to the three-dimensional inviscid Hasegawa-Mima equations

General class of nonlinear bifurcation problems from a point in the essential spectrum: application to shock wave solutions of kinetic equations

An abstract class of bifurcation problems is investigated from the essential spectrum of the associated Frechet derivative. This class is a very general framework for the theory of one-dimensional, steady-profile traveling- shock-wave solutions to a wide family of kinetic integro-differential equations from nonequilibrium statistical mechanics. Such integro-differential equations usually admit the Navier--Stokes system of compressible gas dynamics or the MHD systems in plasma dynamics as a singular limit, and exhibit similar viscous shock layer solutions. The mathematical methods associated with systems of partial differential equations must, however, be replaced by the considerably more complex Bifurcation Theory setting. A hierarchy of bifurcation problems is considered, starting with a simple bifurcation problem from a simple eigenvalue. Sections are entitled as follows: introduction and background from mechanics; the mathematical problem: principal results; a generalized operational calculus, and the derivation of the generalized Lyapunov--Schmidt equations; and methods of solution for the Lyapunov--Schmidt and the functional differential equations. 1 figure. (RWR

The Kuramoto-Sivashinsky equation: Spatio-temporal chaos and intermittencies for a dynamical system

We survey some recent results on the finite-dimensional behavior of the Kuramoto-Sivashinsky equation. We outline how it is rigorously equivalent to a finite dimensional dynamical system on a finite ''inertial'' manifold; a geometric approach to the construction of such a manifold is given. We give some examples of computational simulations supporting the evidence for a low-dimensional vector field which rules the bifurcations of the inertial manifold. 45 refs., 21 figs

Remarks on the Kuramoto-Sivashinsky equation

We report here a joint work in progress on the Kuramoto-Sivashinsky equation. The question we address is the analytical study of a fourth order nonlinear evolution equation. This equation has been obtained by Sivashinsky in the context of combustion and independently by Kuramoto in the context of reaction diffusion-systems. Both were motivated by (nonlinear) stability of travelling waves. Numerical calculations have been done on this equation. All the results seem to indicate a chaotic behavior of the solution. Therefore, the analytical study is of interest in analogy with the Burger's and Navier-Stokes equations. Here we give some existence and uniqueness results for the equation in space dimension one, and we also study a fractional step method of numerical resolution. In a forthcoming joint paper with R. Temam, we will study the asymptotic behavior, as t approaches infinity, of the solution of (0.1) and give an estimate on the number of determining modes